Problem: Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 3,$ $\|\mathbf{w}\| = 7,$ and $\mathbf{v} \cdot \mathbf{w} = 10,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$
Solution: Note that
\begin{align*}
\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| &= \left\| \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w} \right\| \\
&= \frac{|\mathbf{v} \cdot \mathbf{w}|}{\|\mathbf{w}\|^2} \cdot \|\mathbf{w}\| \\
&= \frac{|\mathbf{v} \cdot \mathbf{w}|}{\|\mathbf{w}\|} \\
&= \boxed{\frac{10}{7}}.
\end{align*}